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Properties

Label 1-1287-1287.50-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.725 + 0.688i$
Analytic cond. $138.307$
Root an. cond. $138.307$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.994 + 0.104i)5-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.5 + 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (0.104 − 0.994i)17-s + (−0.743 − 0.669i)19-s + (−0.587 − 0.809i)20-s + 23-s + (0.978 + 0.207i)25-s + (−0.406 − 0.913i)28-s + (−0.978 + 0.207i)29-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.994 + 0.104i)5-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.5 + 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (0.104 − 0.994i)17-s + (−0.743 − 0.669i)19-s + (−0.587 − 0.809i)20-s + 23-s + (0.978 + 0.207i)25-s + (−0.406 − 0.913i)28-s + (−0.978 + 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.725 + 0.688i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.725 + 0.688i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.265463153 + 0.9031886747i\)
\(L(\frac12)\) \(\approx\) \(2.265463153 + 0.9031886747i\)
\(L(1)\) \(\approx\) \(1.095946944 + 0.4087188190i\)
\(L(1)\) \(\approx\) \(1.095946944 + 0.4087188190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
5 \( 1 + (0.994 + 0.104i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (-0.406 + 0.913i)T \)
37 \( 1 + (0.743 - 0.669i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.743 + 0.669i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.994 + 0.104i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.70540977391200599049871029809, −20.23005444629263440961675686761, −19.03648374880959595236422749445, −18.57089510168361273171350556993, −17.68145159891230925548982037822, −16.962157410921392277767097009295, −16.8093295441366178952703686856, −15.11458870991977051444058312303, −14.44137488477059800025839424921, −13.591462952230516503710644292938, −12.93921893778398085194262435248, −12.222659518959522985625644529118, −11.047956273488794595829619588263, −10.74816962633877986799487477225, −9.814765965202199677119615346978, −9.0816484796121406800987963383, −8.25327781024388038973389011981, −7.538133053889517970491516818911, −6.28060043974689497391057190441, −5.30371382441553932777312428079, −4.43950636824283313578018127291, −3.56323165833156545922523212025, −2.29234120793727066811039964527, −1.721666561615545050762717272379, −0.8244784532482786902998034399, 0.71761512816280057803565272766, 1.71402216510395324961395716773, 2.6531595969663459798331582785, 4.22793995972615317558062289278, 5.263439374044114253006691104820, 5.49975236534411790380614007620, 6.77028380523870661810504990695, 7.23855678936204230720659488242, 8.41426911338655731535508870193, 9.00966142854880687525967541964, 9.69015550787001879788690561120, 10.70674802146167234352639532314, 11.27958969508605824479168915341, 12.6508556566667531863036999728, 13.449182992916330849432146917740, 14.163674376295319476824260696994, 14.81803855406592759897308496238, 15.4290584694427473270340131438, 16.584899920766522331084096446447, 17.03437950875293726953360060873, 18.00852801433040225488604803095, 18.18176848790126267016672458140, 19.11231042604546948646444757349, 20.12047445397307420840540364717, 21.03220837887230952978109656692

Graph of the $Z$-function along the critical line